Class XII Mathematics

Chapter 3: Matrices and Determinants

Standard NCERT & CBSE aligned study curriculum. Master concepts, track accuracy, revise weak areas, and challenge yourself with 9 customized practice modes.

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Chapter Overview

Welcome to Class XII Mathematics: Matrices and Determinants. This chapter forms a core structural component of the math syllabus, designed to build analytical rigor and key formula models.

Use the detailed subtopic guide below to review standard definitions, key mathematical rules, and study guidelines.

Prerequisite Concepts

Linear Equations in Two VariablesPair of Linear Equations

About This Chapter

This comprehensive study guide for Matrices and Determinants is designed for Class XII students following the CBSE and NCERT Mathematics curriculum. It covers 5 key subtopics including Matrix operations and properties, Transpose, symmetric, and skew-symmetric matrices, Determinant of square matrix, and 2 more essential concepts. Whether you are preparing for school examinations, CBSE board exams, or competitive tests, this resource provides everything you need to build a strong conceptual foundation and achieve mastery.

The chapter includes 2 key formulas and equations, 1 fully worked step-by-step example problems, interactive practice exercises across 9 difficulty categories, timed mock quizzes, and downloadable worksheets. Each topic is explained with detailed concept definitions, mathematical representations, and expert study guidelines to help you understand not just the "how" but the "why" behind every formula and method.

Mathematics is a subject that rewards consistent practice and conceptual clarity over rote memorization. As you work through this chapter on Matrices and Determinants, focus on understanding the underlying principles first, then gradually increase problem difficulty. Use the practice sections to identify and strengthen weak areas, and refer to the common mistakes section to avoid the pitfalls that most students encounter.

What You'll Learn in This Chapter

By the end of studying Matrices and Determinants for Class XII, you will have developed proficiency in the following learning outcomes as outlined by the NCERT syllabus:

Multiply multi-dimensional matrices.

Calculate adjoints and inverses of 3x3 square matrices.

Solve system of 3 linear equations using Cramer's / Matrix inversion method.

Prerequisites for This Chapter

Before studying Matrices and Determinants, make sure you are comfortable with the following prerequisite concepts. A strong foundation in these areas will help you understand new topics faster and solve problems more confidently:

Linear Equations in Two VariablesPair of Linear Equations

If any of these prerequisites feel unfamiliar, consider reviewing them first using the Related Chapters section at the bottom of this page. Building a solid base ensures you can tackle Matrices and Determinants with full confidence.

Real-World Applications of Matrices and Determinants

Students often wonder “Where will I use Matrices and Determinantsin real life?” The answer is: everywhere. The mathematical concepts you learn in this chapter have practical applications across science, engineering, technology, medicine, finance, and everyday problem-solving. Here are some notable examples:

Computer Graphics Transformations

Every rotation, scaling, and translation in 3D graphics is performed using matrix multiplication — the backbone of game engines.

Google PageRank Algorithm

Google's search ranking uses matrix operations on a massive link graph to determine the importance of every webpage on the internet.

Machine Learning & Neural Networks

Neural network computations are essentially matrix multiplications — forward propagation and backpropagation rely on matrix algebra.

Economics — Input-Output Models

Leontief's input-output model uses matrices to analyze interdependencies between different sectors of an economy.

Understanding the real-world relevance of Matrices and Determinants not only makes learning more engaging but also helps you appreciate how mathematical thinking is a superpower that opens doors in virtually every career path — from engineering and medicine to finance and technology.

Study Tips for Matrices and Determinants

Follow these expert study strategies to maximize your understanding and exam performance in this chapter. These tips are specifically tailored for the type of content covered in Matrices and Determinants:

✏️

Practice Step-by-Step

Write out every intermediate step when solving problems. Skipping steps is the most common source of errors in calculation-heavy chapters. Build speed only after achieving consistent accuracy.

🔄

Verify by Back-Substitution

After finding your answer, substitute it back into the original equation to verify correctness. This simple habit catches most arithmetic and sign errors before they cost you marks.

📝

Maintain an Error Log

Keep a dedicated notebook of mistakes you make during practice. Review it weekly to identify patterns — you will notice the same types of errors recurring and can actively work to eliminate them.

Pro Tip: Consistency beats intensity. Studying Matrices and Determinants for 30 minutes daily is far more effective than cramming for 5 hours before the exam. Use the practice sections below to build muscle memory through regular problem-solving.

Detailed Topic Breakdown

Detailed Subtopics Study Guide

Review detailed conceptual explanations, mathematical equations, and guidelines for each subtopic in this chapter:

1Matrix operations and properties

Concept Explanation

A matrix is a rectangular grid array of elements. Operations include addition (add corresponding elements), scalar multiplication, and matrix multiplication (row-by-column product). Matrix multiplication is not commutative.

Mathematical Representation

(AB)_{ij} = \sum_{k} a_{ik} b_{kj}, \quad AB \neq BA \text{ in general}

Study Guideline: For matrix multiplication AB, the number of columns in A must equal the number of rows in B. The resulting matrix has size (rows of A) x (columns of B).

2Transpose, symmetric, and skew-symmetric matrices

Concept Explanation

The transpose Aᵀ is formed by swapping rows and columns. A square matrix is Symmetric if Aᵀ = A, and Skew-Symmetric if Aᵀ = -A. The diagonal elements of a skew-symmetric matrix are always 0.

Mathematical Representation

(A^T)_{ij} = A_{ji}, \quad A^T = A \, (\text{Symmetric}), \quad A^T = -A \, (\text{Skew-Symmetric})

Study Guideline: Every square matrix can be uniquely expressed as the sum of a symmetric matrix 0.5*(A+Aᵀ) and a skew-symmetric matrix 0.5*(A-Aᵀ).

3Determinant of square matrix

Concept Explanation

A determinant is a scalar value calculated from a square matrix that captures key geometric properties. A matrix is invertible if and only if its determinant is non-zero (non-singular).

Mathematical Representation

\det(A) = |A|, \quad \det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

Study Guideline: Expand determinants along the row or column with the most zeroes to minimize calculation steps.

4Adjoint and inverse of matrix

Concept Explanation

The adjoint (adj A) is the transpose of the cofactor matrix. The inverse matrix A⁻¹ satisfies A * A⁻¹ = I, and is calculated using the adjoint and determinant.

Mathematical Representation

A^{-1} = \frac{1}{|A|} \text{adj}(A) \quad \text{where } |A| \neq 0

Study Guideline: If the determinant |A| is 0, the matrix is singular and does not have an inverse. Verify determinant before calculating the adjoint.

5Solving system of linear equations using matrix inverse

Concept Explanation

A system of linear equations can be represented as AX = B. If A is non-singular, the system has a unique solution calculated by multiplying both sides by the inverse of A.

Mathematical Representation

A X = B \implies X = A^{-1} B \quad \text{where } |A| \neq 0

Study Guideline: This method is known as Matrix Method. Calculate det(A), then adjoint(A), then inverse(A), and multiply by vector B.

Matrix Calculator (2x2 Matrix)

Calculate determinants, trace values, and inverse matrices for any 2x2 mathematical matrix grid.

Live Solver

Adjust Inputs

Matrix [1,1]4

Matrix [1,2]3

Matrix [2,1]2

Matrix [2,2]2

Calculated Results

Determinant (Det)

Inverse Matrix

[[1, -1.5], [-1, 2]]

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NEXPROTOOLS COMPUTATIONAL REPORT

Deterministic Mathematical Simulation Engine • Verified Calculations

Date: 9/1/2026

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Matrix Calculator (2x2 Matrix)

Calculate determinants, trace values, and inverse matrices for any 2x2 mathematical matrix grid.

1. Input Parameters

Parameter	Value	Unit
Matrix [1,1]	4	—
Matrix [1,2]	3	—
Matrix [2,1]	2	—
Matrix [2,2]	2	—

2. Calculated Results

Metric	Calculated Output
Determinant (Det)	2
Inverse Matrix	[[1, -1.5], [-1, 2]]

Generated by NexProTools (www.nexprotools.com)

100% Deterministic Client-Side Simulation

Student Solver & Visualizer Guide

Real-time calculations

Step-by-step solving

1. Collect input parameters:

- a11 = 4

- a12 = 3

- a21 = 2

- a22 = 2

2. Feed parameters through standard algorithmic calculation methods.

3. Compile calculated results:

- determinant = 2

- inverseMatrix = [[1, -1.5], [-1, 2]]

Student-friendly explanations

"This math utility processes input parameters using standard algebraic algorithms. By substituting variables into target formulas, the engine solves equations and computes results instantly with high arithmetic precision!"

Visual explanations

MATH SOLVER RUNNING:
 [Inputs] ──► [Mathematical Formula] ──► [Outputs]
  Processed elements successfully.

Class XII CBSE Board Practice Questions

Review and solve 37 real Board paper questions from the 16 mock sample sets matching Matrices and Determinants.

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Sample Paper 01 | Question 2Section A (1 Mark)

For any square matrix A, AAT is a

Sample Paper 01 | Question 17Section A (1 Mark)

If A is square matrix such that A2 = I , then (A - I)3 + (A + I)2 - 7A is equal to

Sample Paper 01 | Question 18Section A (1 Mark)

If 1 2 1 2 3 1 3 1 a [ ] [ ] [ ] [ ][ ] is non-singular matrix and a ∈ A, then the set A is:

Sample Paper 02 | Question 2Section A (1 Mark)

If A is a square matrix of order n then |jda(A)| =

Sample Paper 02 | Question 5Section A (1 Mark)

If |A| = |kA|, where A is a square matrix of order 2, then sum of all possible values of k is:

Sample Paper 03 | Question 10Section A (1 Mark)

If A is a non-singular matrix of order 3 and |A| = - 4, find |adj A|.

Sample Paper 03 | Question 13Section A (1 Mark)

For two matrices A and B, given that 1 1 4A B- = , then inverse of (4A) is :

Sample Paper 03 | Question 18Section A (1 Mark)

If X, Y and XY are matrices of order 2 х 3, m х n and 2 х 5 respectively, then number of elements in matrix Y is :

Sample Paper 04 | Question 5Section A (1 Mark)

If A is a square matrix of order 3 such that |A| = - 5 , then value of is

Sample Paper 04 | Question 31Section C (3 Marks)

(a) Find a matrix A such that 4 0 17 10 1 2 0 16A [ ] [ ] =[ ] [ ]- - -[ ] [ ] Also, find A-1.

Sample Paper 05 | Question 1Section A (1 Mark)

If A is a square matrix such that A2 = A, then (I + A)2 - 3A is

Sample Paper 05 | Question 6Section A (1 Mark)

If A and B are square matrices of order 3 such that |A| = 1 and |B| = 3, then the value of |3AB| is:

Sample Paper 05 | Question 11Section A (1 Mark)

If A = 1 1 0 x[ ] [ ][ ] and A2 is the identity matrix, then x is equal to:

Sample Paper 06 | Question 1Section A (1 Mark)

Let A be a square matrix of order 2 × 2, then |KA| is equal to

Sample Paper 06 | Question 6Section A (1 Mark)

The sum of cofactors of 7 and 12 in the determinant 1 2 4 5 7 8 9 10 12 is:

Sample Paper 07 | Question 11Section A (1 Mark)

If A and B are square matrices of order 3 such that |A| = 1 and |B| = 3, then the value of |3AB| is:

Sample Paper 07 | Question 16Section A (1 Mark)

The sum of cofactors of 7 and 12 in the determinant 1 2 4 5 7 8 9 10 12 is:

Sample Paper 08 | Question 15Section A (1 Mark)

The inverse of the matrix 1 1 2 3 [ - ] [ ][ ] is:

Sample Paper 09 | Question 4Section A (1 Mark)

The cofactor of (-1) in the matrix 1 0 4 3 5 1 0 1 2 [ ] -[ ] [ ] [ ][ ] is:

Sample Paper 09 | Question 16Section A (1 Mark)

A and B are invertible matrices of the same order such that |(AB)-1| = 8, If |A| = 2, then |B| is

Sample Paper 10 | Question 12Section A (1 Mark)

If A and B are invertible matrices, then which of the following is not correct.

Sample Paper 10 | Question 15Section A (1 Mark)

Let A be a non-singular matrix of order (3 × 3). Then |adj.A| is equal to

Sample Paper 11 | Question 8Section A (1 Mark)

If A is a 3 x 3 matrix and |A| = - 2 then value of |A(adjA)| is

Sample Paper 11 | Question 11Section A (1 Mark)

For any matrix A = [aij], if cij denotes its cofactors then find the value of a11c12 + a12c22 + a13c32.

Sample Paper 11 | Question 15Section A (1 Mark)

If A and B are square matrices of order 3 such that |A| = 1 and |B| = 3, then the value of |3AB| is:

Sample Paper 11 | Question 35Section D (5 Marks)

If A = 2 3 5 3 2 4 1 1 2 -[ ] [ ] -[ ] [ ] -[ ] , find A-1 and hence solve the system of linear equations: 2x - 3y + 5z = 11, 3x + 2y - 4z = -5; x + y - 2z = -3.

Sample Paper 12 | Question 15Section A (1 Mark)

If A is a square matrix of order 3, such that A(adjA) = 10 then is equal to

Sample Paper 13 | Question 34Section D (5 Marks)

The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.

Sample Paper 13 | Question 38Section E (4 Marks)

Case-Study 3: Two schools A and B want to award their selected students on the values of Honesty, Hard work and Punctuality. The school A wants to award ₹ x each, ₹ y each and ₹ z each for the three respective values to its 3, 2 and 1 students respectively with a total award money of ₹ 2200. School B wants to spend ₹ 3100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school A). The total amount of award for one prize on each value is ₹ 1200. Using the concept of matrices and determinants,

Sample Paper 14 | Question 2Section A (1 Mark)

If A and B are invertible matrices, then which of the following is not correct?

Sample Paper 14 | Question 3Section A (1 Mark)

If Cij denotes the cofactor of element Pij of the matrix P = 1 1 2 0 2 3 3 2 4 -[ ] [ ] -[ ] [ ][ ] , then the value of C31.C23 is:

Sample Paper 15 | Question 14Section A (1 Mark)

If for a square matrix A, A² - A + I = O, then A - 1 equals :

Sample Paper 15 | Question 15Section A (1 Mark)

If A is a square matrix of order 3 and |A| = -5, then |adj A| is:

Sample Paper 15 | Question 16Section A (1 Mark)

If A = 2 6 1 1 x[ ] [ ]-[ ] is a singular matrix, then x :

Sample Paper 15 | Question 38Section E (4 Marks)

Case-Study 3: Three car dealers, say A, B and C, deals in three types of cars, namely Hatchback cars, Sedan cars, SUV cars. The sales figure of 2019 and 2020 showed that dealer A sold 120 Hatchback, 50 Sedan, 10 SUV cars in 2019 and 300 Hatchback, 150 Sedan, 20 SUV cars in 2020; dealer B sold 100 Hatchback, 30 Sedan, 5 SUV cars in 2019 and 200 Hatchback, 50 Sedan, 6 SUV cars in 2020; dealer C sold 90 Hatchback, 40 Sedan, 2 SUV cars in 2019 and 100 Hatchback, 60 Sedan, 5 SUV cars in 2020. (i) Write the matrix summarizing sales data of 2019 and 2020. (ii) Find the matrix summarizing sales data of 2020. (iii) Find the total number of cars sold in two given years, by each dealer? OR (iii) If each dealer receives a profit of = 50000 on sale of a Hatchback, 100000 on sale of a Sedan and 2200000 on sale of an SUV, then find the amount of p rofit received in the year 2020 by each dealer.

Sample Paper 16 | Question 10Section A (1 Mark)

If A = 1 1 0 x[ ] [ ][ ] and A2 is the identity matrix, then x is equal to:

Sample Paper 16 | Question 12Section A (1 Mark)

If A is a square matrix of order 3 and |A| = 5, then |Ajda| =

Print-Ready Math Worksheet

Generate and print a beautiful chapter worksheet complete with key equations, problems, and dedicated writing spaces.

Worksheet Details:

• Title: Class XII Math: Matrices and Determinants Printable Sheet
• Format: Print Optimized CSS stylesheet
• Sections: Formulas checklist, 5 assessment questions, worked keys

NexPro Education Worksheet

Class XII Mathematics: Matrices and Determinants

Name: ______________________Date: __________________Score: ______ / 5

Section I: Key Formulas

Matrix Inverse Formula

A^{-1} = \frac{1}{|A|} \text{adj}(A)

Determinant Expansion 2x2

\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc